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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Hilbert's irreducibility theorem, conceived by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
in 1892, states that every finite set of
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
s in a finite number of variables and having
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.


Formulation of the theorem

Hilbert's irreducibility theorem. Let :f_1(X_1, \ldots, X_r, Y_1, \ldots, Y_s), \ldots, f_n(X_1, \ldots, X_r, Y_1, \ldots, Y_s) be irreducible polynomials in the ring :\Q(X_1, \ldots, X_r) _1, \ldots, Y_s Then there exists an ''r''-tuple of rational numbers (''a''1, ..., ''ar'') such that :f_1(a_1, \ldots, a_r, Y_1,\ldots, Y_s), \ldots, f_n(a_1, \ldots, a_r, Y_1,\ldots, Y_s) are irreducible in the ring :\Q _1,\ldots, Y_s Remarks. * It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is Zariski dense in \Q^r. * There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''1, ..., ''ar'') to be integers. * There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, number fields are Hilbertian.Lang (1997) p.41 * The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take n=r=s=1 in the definition. A result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of n=r=s=1 and f=f_1 absolutely irreducible, that is, irreducible in the ring ''K''alg 'X'',''Y'' where ''K''alg is the algebraic closure of ''K''.


Applications

Hilbert's irreducibility theorem has numerous applications in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. For example: * The inverse Galois problem, Hilbert's original motivation. The theorem almost immediately implies that if a finite group ''G'' can be realized as the Galois group of a Galois extension ''N'' of ::E=\Q(X_1, \ldots, X_r), :then it can be specialized to a Galois extension ''N''0 of the rational numbers with ''G'' as its Galois group.Lang (1997) p.42 (To see this, choose a monic irreducible polynomial ''f''(''X''1, ..., ''Xn'', ''Y'') whose root generates ''N'' over ''E''. If ''f''(''a''1, ..., ''an'', ''Y'') is irreducible for some ''ai'', then a root of it will generate the asserted ''N''0.) * Construction of elliptic curves with large rank. * Hilbert's irreducibility theorem is used as a step in the
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...
proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
. * If a polynomial g(x) \in \Z /math> is a perfect square for all large integer values of ''x'', then ''g(x)'' is the square of a polynomial in \Z This follows from Hilbert's irreducibility theorem with n=r=s=1 and ::f_1(X, Y) = Y^2 - g(X). :(More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.


Generalizations

It has been reformulated and generalized extensively, by using the language of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. See thin set (Serre).


References

* D. Hilbert, "Uber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligen Coefficienten", J. reine angew. Math. 110 (1892) 104–129. *{{cite book, first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine Geometry , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 *J. P. Serre, ''Lectures on The Mordell-Weil Theorem'', Vieweg, 1989. *M. D. Fried and M. Jarden, ''Field Arithmetic'', Springer-Verlag, Berlin, 2005. *H. Völklein, ''Groups as Galois Groups'', Cambridge University Press, 1996. *G. Malle and B. H. Matzat, ''Inverse Galois Theory'', Springer, 1999. Theorems in number theory Theorems about polynomials David Hilbert